In summary, we first apply the irrelevant technique to \(G\)  in \(2^{O(k)}n\)  time
so that \(G\)  has treewidth of \(2^{O(k)}\)  in Section .
We construct the frames, skeleton forests, and reference paths in \(2^{O(k)}n\)  time in Section  and Section .
Using them, we enumerate \(2^{O(k^2)}\)  weak \(T\) -linkages satisfying the properties stated in Lemma REF 
in \(2^{O(k^2)}n\)  time.
Then for each weak \(T\) -linkage, we apply the algorithm of [1]} after modifying \(G\)  slightly
in \(2^{O(k)}n\)  time in total as described in Section .
In this way, we can solve the Planar Disjoint Paths problem in \(2^{O(k^2)}n\)  time.
